Make sure the triangle you are given is right-angled, as the Pythagorean theorem only applies to right-angled triangles. In right-angled triangles, one of the three angles is always 90 degrees.
- A right angle in a right triangle is indicated by a square icon, not a curve, which is an oblique angle.
Add guidelines for the sides of the triangle. Label the legs as "a" and "b" (legs - sides intersecting at right angles), and the hypotenuse as "c" (hypotenuse - the largest side right triangle lying opposite right angle).
Determine which side of the triangle you want to find. The Pythagorean theorem allows you to find any side of a right triangle (if the other two sides are known). Determine which side (a, b, c) you need to find.
- For example, given a hypotenuse equal to 5, and given a leg equal to 3. In this case, you need to find the second leg. We'll come back to this example later.
- If the other two sides are unknown, it is necessary to find the length of one of the unknown sides in order to be able to apply the Pythagorean theorem. To do this, use the basic trigonometric functions(if you are given the value of one of the oblique angles).
Substitute in the formula a 2 + b 2 = c 2 the values given to you (or the values you found). Remember that a and b are legs and c is hypotenuse.
- In our example, write: 3² + b² = 5².
Square each side you know. Or leave the degrees - you can square the numbers later.
- In our example, write: 9 + b² = 25.
Isolate the unknown side on one side of the equation. To do this, transfer the known values to the other side of the equation. If you find the hypotenuse, then in the Pythagorean theorem it is already isolated on one side of the equation (so nothing needs to be done).
- In our example, move 9 to the right side of the equation to isolate the unknown b². You will get b² = 16.
Retrieve Square root from both sides of the equation. At this stage, there is an unknown (squared) on one side of the equation, and an intercept (number) on the other side.
- In our example, b² = 16. Take the square root of both sides of the equation and get b = 4. So the second leg is 4 .
Use the Pythagorean theorem in Everyday life, since it can be used in a large number practical situations. To do this, learn to recognize right-angled triangles in everyday life - in any situation in which two objects (or lines) intersect at right angles, and a third object (or line) connects (diagonally) the tops of the first two objects (or lines), you can use the Pythagorean theorem to find the unknown side (if the other two sides are known).
- Example: given a staircase leaning against a building. The bottom of the stairs is 5 meters from the base of the wall. The top of the stairs is 20 meters from the ground (up the wall). How long are the stairs?
- "5 meters from the base of the wall" means that a = 5; "Is 20 meters from the ground" means that b = 20 (that is, you are given two legs of a right-angled triangle, since the wall of the building and the surface of the Earth intersect at right angles). The length of the ladder is the length of the hypotenuse, which is unknown.
- a² + b² = c²
- (5) ² + (20) ² = c²
- 25 + 400 = c²
- 425 = c²
- c = √425
- s = 20.6. So the approximate length of the ladder is 20.6 meters.
- "5 meters from the base of the wall" means that a = 5; "Is 20 meters from the ground" means that b = 20 (that is, you are given two legs of a right-angled triangle, since the wall of the building and the surface of the Earth intersect at right angles). The length of the ladder is the length of the hypotenuse, which is unknown.
Pythagorean theorem Is one of the fundamental theorems of Euclidean geometry, establishing the relation
between the sides of a right-angled triangle.
It is believed to have been proven by the Greek mathematician Pythagoras, after whom it was named.
Geometric formulation of the Pythagorean theorem.
Initially, the theorem was formulated as follows:
In a right-angled triangle, the area of the square built on the hypotenuse is equal to the sum of the areas of the squares,
built on legs.
Algebraic formulation of the Pythagorean theorem.
In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
That is, denoting the length of the hypotenuse of a triangle by c, and the lengths of the legs through a and b:
Both formulations Pythagorean theorems are equivalent, but the second formulation is more elementary, it is not
requires the concept of area. That is, the second statement can be verified without knowing anything about the area and
by measuring only the lengths of the sides of a right-angled triangle.
The converse theorem of Pythagoras.
If the square of one side of the triangle is equal to the sum of the squares of the other two sides, then
rectangular triangle.
Or, in other words:
For any triple of positive numbers a, b and c such that
there is a right-angled triangle with legs a and b and hypotenuse c.
Pythagoras' theorem for an isosceles triangle.
Pythagoras' theorem for an equilateral triangle.
Proofs of the Pythagorean theorem.
At the moment, 367 proofs of this theorem have been recorded in the scientific literature. Probably the theorem
Pythagoras is the only theorem with such an impressive number of proofs. Such diversity
can only be explained by the fundamental meaning of the theorem for geometry.
Of course, conceptually all of them can be divided into a small number of classes. The most famous of them:
proof area method, axiomatic and exotic evidence(for example,
by using differential equations).
1. Proof of the Pythagorean theorem through similar triangles.
The following proof of the algebraic formulation is the simplest of the proofs under construction
directly from the axioms. In particular, it does not use the concept of the area of a figure.
Let be ABC there is a right-angled triangle with a right angle C... Let's draw the height from C and denote
its foundation through H.
Triangle ACH like a triangle AB C in two corners. Similarly, triangle CBH is similar ABC.
Introducing the notation:
we get:
,
which corresponds to -
By adding a 2 and b 2, we get:
or, as required to prove.
2. Proof of the Pythagorean theorem by the area method.
The proofs below, despite their seeming simplicity, are not at all so simple. All of them
use the properties of the area, the proof of which is more difficult than the proof of the Pythagorean theorem itself.
- Proof through equal complementarity.
Place four equal rectangular
triangle as shown in the figure
on right.
Quadrilateral with sides c- square,
since the sum of two acute angles is 90 °, and
expanded angle - 180 °.
The area of the entire figure is, on the one hand,
area of a square with side ( a + b), and on the other hand, the sum of the areas of the four triangles and
Q.E.D.
3. Proof of the Pythagorean theorem by the method of infinitesimal.
Considering the drawing shown in the figure, and
watching the side changea, we can
write the following relation for infinitely
small side incrementswith and a(using the similarity
triangles):
Using the variable separation method, we find:
More general expression to change the hypotenuse in case of increments of both legs:
Integrating this equation and using the initial conditions, we get:
Thus, we arrive at the desired answer:
As it is easy to see, the quadratic dependence in the final formula appears due to the linear
proportionality between the sides of the triangle and the increments, while the sum is related to independent
contributions from the increment of different legs.
A simpler proof can be obtained if we assume that one of the legs does not experience an increment
(in this case, the leg b). Then for the constant of integration we get:
Ministry of Education and Science of the Russian Federation
Municipal educational institution
Leboterskaya main comprehensive school
Chainsky district Tomsk region
ESSAY
on this topic: Pythagoras and his theorem
Completed:
8th grade students
Pchelkina Irina
Makarova Nadezhda
Supervisor:
V.K. Stasenko,
mathematic teacher
Introduction ………………………………… .. …………………………………… .. 3
1. From the biography of Pythagoras ……………………………………………… ..3
2. Pythagoras and the Pythagoreans …………………………………………………. …4
3. From the history of the creation of the theorem ………………………………………… .. ..5
4. Six proofs of the theorem …………………………………………… .6
4.1. Ancient Chinese evidence ……………………………………… 6
4.2. Proof of J. Gardfield ……………………………………… 7.
4.3 The oldest evidence ………………………………………… .. 8.
4.4. The simplest proof …………………………………………… 9
4.5 Proof of the Ancients ……………………………………………… 10
4.6. Euclid's proof …………………………………………… ..11.
5. Application of the Pythagorean theorem …………………………………………… 12
5.1. Theoretical tasks ……………………………………………… ..13
5.2. Practical tasks (old) …………………………………… 14
Conclusion ……………………………………………………………………… 15
References …………………………………………………………… 16
INTRODUCTION
In that academic year we got acquainted with an interesting theorem, known, as it turned out from ancient times:
"The square built on the hypotenuse of a right-angled triangle is equal to the sum of the squares built on the legs."
Usually the discovery of this statement is attributed to the ancient Greek philosopher and mathematician Pythagoras (VI century BC). But the study of ancient manuscripts showed that this statement was known long before the birth of Pythagoras.
We wondered why, in this case, it is associated with the name of Pythagoras.
The purpose of our research was: to find out who Pythagoras was and what relation he has to this theorem. Studying the history of the theorem, we decided to find out:
o Are there other proofs of this theorem?
o What is the significance of this theorem in people's lives?
o What role did Pythagoras play in the development of mathematics?
1. From the biography of Pythagoras
Pythagoras of Samos is a great Greek scientist. His name is familiar to every student. If asked to name one ancient mathematician, the absolute majority will name Pythagoras. Its fame is associated with the name of the Pythagorean theorem. Although now we already know that this theorem was known in ancient Babylon 1200 years before Pythagoras, and in Egypt 2000 years before it was known a right-angled triangle with sides 3, 4, 5, we still call it by the name of this ancient scientist.
Almost nothing is reliably known about the life of Pythagoras, but it is connected with his name a large number of legends.
Pythagoras was born in 570 BC. e on the island of Samos. The father of Pythagoras was Mnesarchus, a gem cutter. Mnesarchus, according to Apuleius, "was famous among the masters for his art of carving gems," but acquired fame rather than wealth. The name of Pythagoras's mother has not survived.
Pythagoras had a beautiful appearance, wore a long beard, and a golden diadem on his head. Pythagoras is not a name, but a nickname that the philosopher received for always speaking correctly and convincingly, like a Greek oracle. (Pythagoras - "convincing by speech".)
Among the teachers of the young Pythagoras were the elder Germodamant and Therekides of Syros (although there is no firm conviction that it was Hermodamantes and Therekides who were the first teachers of Pythagoras). Young Pythagoras spent whole days at the feet of the elder Hermodamantes, listening to the melodies of the cithara and the hexameters of Homer. Pythagoras retained the passion for music and poetry of the great Homer throughout his life. And, being a recognized sage, surrounded by a crowd of disciples, Pythagoras began the day by singing one of Homer's songs.
Ferekid was a philosopher and was considered the founder of the Italian school of philosophy. Thus, if Hermodamantes introduced the young Pythagoras to the circle of muses, then Therekides turned his mind to the Logos. Ferekid directed the gaze of Pythagoras to nature and in it alone he advised to see his first and main teacher.
But be that as it may, the restless imagination of young Pythagoras very soon became cramped on little Samos, and he went to Miletus, where he met another scientist - Thales. Thales advised him to go to Egypt for knowledge, which Pythagoras did.
In 550 BC. e Pythagoras makes a decision and goes to Egypt. So, before Pythagoras, an unknown country and an unknown culture opens up. Much amazed and surprised Pythagoras in this country, and after some observation of the life of the Egyptians, Pythagoras realized that the path to knowledge protected by the caste of priests lies through religion.
Together with the Egyptian boys, he sat down at the limestone plates, a mature Ellen with a black curly beard. But unlike their smaller companions, the bearded Ellin's ears were not on his back, and his head was still. Very soon, Pythagoras far outstripped his classmates. But the school of scribes was only the first step on the path to secret knowledge.
After eleven years of study in Egypt, Pythagoras goes home, where on the way he falls into Babylonian captivity. There he became acquainted with Babylonian science, which was more developed than Egyptian. The Babylonians were able to solve linear, quadratic, and some types of cubic equations. They successfully applied the Pythagorean theorem more than 1000 years before Pythagoras. After escaping from captivity, he could not stay in his homeland for a long time because of the atmosphere of violence and tyranny that reigned there. He decided to move to Croton (Greek colony in northern Italy).
It is in Croton that the most glorious period in the life of Pythagoras begins. There he established something like a religious-ethical brotherhood or a secret monastic order, whose members pledged to lead the so-called Pythagorean way of life.
2. Pythagoras and the Pythagoreans
Pythagoras organized in the Greek colony in the south of the Apennine Peninsula a religious and ethical brotherhood, such as a monastic order, which would later be called the Pythagorean Union. The members of the union had to adhere to certain principles: firstly, to strive for the beautiful and glorious, secondly, to be useful, and thirdly, to strive for high pleasure.
The system of moral and ethical rules, bequeathed by Pythagoras to his students, was collected in a kind of moral code of the Pythagoreans "Golden
poems ", which were very popular in the era of Antiquity, the Middle Ages and the Renaissance. The Pythagorean system of studies consisted of three sections:
· Teachings about numbers - arithmetic,
· Doctrines about figures - geometry,
· The doctrine of the structure of the Universe - astronomy.
The educational system laid down by Pythagoras existed for many centuries.
The Pythagoreans taught that God put numbers at the heart of the world order. God is unity, and the world is many and consists of opposites. That which brings opposites to unity and unites everything into space is harmony. Harmony is divine and is expressed in numbers. Whoever studies harmony to the end will himself become divine and immortal.
Music, harmony and numbers were inextricably linked in the teachings of the Pythagoreans. Mathematics and numerical mysticism were fantastically mixed in it. Pythagoras believed that number is the essence of all things and that the universe is harmonic system numbers and their relationships.
The Pythagorean school did a lot to give geometry the character of a science. The main feature of the Pythagorean method was the combination of geometry with arithmetic.
Pythagoras dealt a lot with proportions and progressions and, probably, the similarity of figures, since he is credited with solving the problem: "Using these two figures, construct a third, equal in size to one of the data and similar to the second."
Pythagoras and his students introduced the concept of polygonal, friendly, perfect numbers and studied their properties. Pythagoras was not interested in arithmetic as a practice of calculations, and he proudly declared that "he put arithmetic above the interests of the merchant."
Pythagoras was one of the first to believe that the Earth has the shape of a ball and is the center of the Universe, that the Sun, Moon and planets have their own motion, different from the daily motion of fixed stars.
Nicolaus Copernicus took the Pythagorean doctrine of the Earth's motion as the prehistory of his heliocentric doctrine. It is not for nothing that the Church declared Copernicus' system a "false Pythagorean teaching."
In the school of Pythagoras, the discoveries of the students were attributed to the teacher, so it is almost impossible to determine what Pythagoras himself did and what his students did.
Disputes have been going on around the Pythagorean union for the third millennium, but there is still no general opinion. The Pythagoreans had many symbols and signs, which were a kind of commandments: for example, "do not walk through the scales", i.e. do not violate justice; Do not stir up fire with a knife, ”that is, do not offend angry people with offensive words.
But the main Pythagorean symbol is
a symbol of health and identification mark –
there was a pentagram or a Pythagorean star -
star pentagon formed by diagonals
regular pentagon.
The inhabitants of many cities in Greece were members of the Pythagorean union.
The Pythagoreans also accepted women into their society. The union flourished for more than twenty years, and then persecutions began against its members, many of the disciples were killed.
There were many different legends about the death of Pythagoras himself. But the teachings of Pythagoras and his disciples continued to live.
3. From the history of the Pythagorean theorem
It is now known that this theorem was not discovered by Pythagoras. However, some believe that it was Pythagoras who was the first to give its full-fledged proof, while others deny him this merit. Some attribute to Pythagoras the proof which Euclid gives in the first book of his Principles. On the other hand, Proclus asserts that the proof in the Elements belongs to Euclid himself.
As we can see, the history of mathematics has hardly preserved any reliable concrete data on the life of Pythagoras and his mathematical activities. On the other hand, the legend reports even the immediate circumstances accompanying the discovery of the theorem. Many people know the sonnet of the German novelist Chamisso:
We begin our historical survey of the Pythagorean theorem with ancient China. Here Special attention attracted by the mathematical book Chu-pei. This essay says so about the Pythagorean triangle with sides 3, 4 and 5:
"If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5 when the base is 3 and the height is 4" .
It is very easy to reproduce their way of building. Take a rope 12 m long and tie it to it along a colored strip at a distance of 3 m. from one end and 4 meters from the other.
The right angle will be enclosed between the sides 3 and 4 meters long. In the same book, a drawing is proposed that coincides with one of the drawings of the Hindu geometry of Baskhara.
Cantor(the largest German historian of mathematics) believes that the equality 3 ² + 4 ² = 5 ² was already known to the Egyptians around 2300 BC. e., during the time of King Amenemhat I (according to papyrus 6619 of the Berlin Museum).
According to Cantor, harpedonapts, or "rope pulls", built right angles using right-angled triangles with sides 3, 4, and 5.
The Babylonians knew a little more about the Pythagorean theorem. In one text, dating back to the time of Hammurabi, i.e. by 2000 BC, an approximate calculation of the hypotenuse of a right triangle is given; from this we can conclude that in Mesopotamia they knew how to perform calculations with right-angled triangles, at least in some cases.
Geometry among the Indians was closely associated with the cult. It is very likely that the hypotenuse square theorem was known in India already around the 8th century BC. Along with purely ritual prescriptions, there are also works of a geometrically theological nature, called Sulvasutras. In these writings, dating back to the 4th or 5th century BC, we meet with the construction of a right angle using a triangle with sides 15, 36, 39.
In the Middle Ages the Pythagorean theorem determined the boundary, if not the greatest possible, then at least good mathematical knowledge. A characteristic drawing of the Pythagorean theorem, which now sometimes turns into schoolchildren, for example, into a professor dressed in a robe or a man in a top hat, in those days was often used as a symbol of mathematics.
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In conclusion, we present various formulations of the Pythagorean theorem translated from Greek, Latin and German.
Euclid this theorem reads (literal translation):
"In a right-angled triangle, the square of the side stretched over the right angle is equal to the squares on the sides that enclose the right angle."
Latin translation of arabic text Annaritia(about 900 BC) by Gerhard Kremonsky(12th century) reads (translated):
"In any right-angled triangle, a square formed on a side stretched over a right angle is equal to the sum of two squares formed on two sides that enclose a right angle."
In Geometry Culmonensis (around 1400), the theorem reads like this (translated):
“So, the area of a square, measured along the length of a side, is as large as that of two squares, which are measured on two sides of it, adjacent to a right angle "
In the Russian translation of the Euclidean "Elements", the Pythagorean theorem is stated as follows:
"In a right-angled triangle, the square of the side opposite to the right angle is equal to the sum of the squares of the sides containing the right angle."
As you can see, in different countries and in different languages there are different versions of the formulation of the theorem we are familiar with. Created in different time and in different languages, they reflect the essence of one mathematical pattern, the proof of which also has several options.
4. Six ways to prove the Pythagorean theorem
4.1. Ancient Chinese proof
In an ancient Chinese drawing, four equal right-angled triangles with legs a , b and hypotenuse with stacked so that their outer contour forms a square with a side a + b, and the inner one is a square with side with built on the hypotenuse
a 2 + 2ab + b 2 = c 2 + 2ab
a 2 + b 2 = c 2
4.2. Proof by J. Gardfield (1882)
Place two equal right-angled triangles so that the leg of one of them is a continuation of the other.
The area of the trapezoid under consideration is found as the product of the half-sum of the bases and the height
On the other hand, the area of the trapezoid is equal to the sum of the areas of the resulting triangles:
Equating these expressions, we get:
or with 2 =
a
2
+
b
2
4.3. Oldest proof
(contained in one of the works of Bhaskara).
Let ABCD be a square whose side is equal to the hypotenuse of a right-angled triangle ABE (AB = c, BE = a,
Let CK BE = a, DL CK, AM DL
ΔABE = ∆BCK = ∆CDL = ∆AMD,
means KL = LM = ME = EK = a-b.
4.4. The simplest proof
4.5. Proof of ancient Hindus [ 2]
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A square with a side (a + b) can be divided into parts either as in Figure a) or as in Figure b). It is clear that the parts 1,2,3,4 in both figures are the same. And if we subtract equal from equal (areas), then they will remain equal, i.e. c 2 = a 2 + b 2 .
However, the ancient Hindus, to whom this reasoning belongs, usually did not write it down, but accompanied it with only one word:
Look!
4.6. Euclid's proof
For two millennia, the most common was Euclid's proof of the Pythagorean theorem. It is included in his famous book "Beginnings".
Euclid lowered the height BH from the vertex of the right angle to the hypotenuse and argued that its continuation divides the square completed on the hypotenuse into two rectangles, the areas of which are equal to the areas of the corresponding squares built on the legs.
The drawing used to prove this theorem is jokingly called "Pythagorean pants." For a long time, it was considered one of the symbols of mathematical science.
Pupils of the Middle Ages considered the proof of the Pythagorean theorem very difficult and called it Dons asinorum - donkey bridge, or elefuga - flight of the "poor", since some "poor" students who did not have serious mathematical training fled from geometry. Weak students, who had learned the theorems by heart, without understanding, and therefore called "donkeys", were unable to overcome the Pythagorean theorem, which served as an insurmountable bridge for them. Because of the drawings accompanying the Pythagorean theorem, the students also called it a "windmill", composed poems like "Pythagorean pants are equal on all sides," and drew cartoons.
5. Application of the Pythagorean theorem.
5.1. Theoretical modern tasks
1. The perimeter of the rhombus is 68 cm, and one of its diagonals is 30 cm. Find the length of the other diagonal of the rhombus.
2. The hypotenuse KR of a right-angled triangle KMR is equal to cm, and the leg of MR is 4 cm. Find the median PC.
3. Squares are built on the sides of a right-angled triangle, and
S 1 -S 2 = 112 cm 2 and S 3 = 400 cm 2. Find the perimeter of the triangle.
4. Given a triangle ABC, angle C = 90 0, CD AB, AC = 15 cm, AD = 9 cm.
Find AB.
5.2. Practical old tasks
5. To secure the mast, you need to install
4 ropes. One end of each cable should be fixed at a height of 12 m, the other on the ground at a distance of 5 m from the mast. Will 50 m of cable be enough to secure the mast?
6... The problem of the 12th century Indian mathematician Bhaskara
“A lonely poplar grew on the bank of the river.
Suddenly a gust of wind broke his trunk.
The poor poplar fell. And the angle is straight
With the flow of the river, its trunk was.
Remember now that there is a river in that place
It was only four feet wide.
The top bent at the edge of the river.
There are only three feet left of the trunk,
Please tell me soon now:
How big is the height of the poplar? "
7... Problem from the textbook "Arithmetic" by Leonty Magnitsky [ 19]
"If a certain person has a ladder to the wall, clean up the walls, the same height is 117 feet. And you will find a ladder with a length of 125 feet."
And the Vedati wants to keep the lower end of the stairs from the wall by sowing the stairs with a number of feet. "
8... Problem from Chinese "Mathematics in Nine Books"
"There is a reservoir with a side of 1 zhang = 10 chi. In its center there is a reed growing, which protrudes 1 chi above the water. If you pull the reeds to the shore, it will just touch it.
The question is: what is the depth of the water and how long is the reed? "
Conclusion
The Pythagorean theorem is so famous that it is difficult to imagine a person who has not heard about it. We studied a number of historical and mathematical sources, including information on the Internet, and saw that the Pythagorean theorem is interesting not only for its history, but also because it occupies an important place in life and science. This is evidenced by the various interpretations of the text of this theorem and the ways of its proof given by us in this work.
So, the Pythagorean theorem is one of the main and, one might say, the most important theorem of geometry. Its significance lies in the fact that most of the theorems of geometry can be derived from it or with its help. The Pythagorean theorem is also remarkable in that in itself it is not at all obvious. For example, the properties of an isosceles triangle can be seen directly in the drawing. But no matter how you look at a right-angled triangle, you will never see that there is a simple relationship between its sides: c 2 = a 2 + b 2. Therefore, visualization is often used to prove it.
The merit of Pythagoras was that he gave a full scientific proof of this theorem.
The personality of the scientist himself is interesting, the memory of whom this theorem is not accidentally preserved. Pythagoras is a wonderful orator, teacher and educator, the organizer of his school, focused on the harmony of music and numbers, kindness and justice, on knowledge and healthy image life. He may well serve as an example for us, distant descendants.
Literature and Internet resources:
1. G.I. Glazer History of Mathematics in School VII - VIII grades, a guide for teachers, - M: Enlightenment 1982.
2. I. Ya. Dempan, N. Ya. Vilenkin "Behind the Pages of a Mathematics Textbook" A manual for students in grades 5-6, Moscow, Enlightenment 1989.
3. I.G. Zenkevich "Aesthetics of a lesson in mathematics", Moscow: Enlightenment, 1981.
4. Voytikova N.V. "Pythagorean theorem" course work, Anzhero-Sudzhensk, 1999
5. V. Litzman, The Pythagorean Theorem, Moscow 1960.
6. A.V. Voloshinov "Pythagoras" M. 1993.
7. LF Pichurin "Behind the Pages of the Algebra Textbook" M. 1990.
8. A. N. Zemlyakov "Geometry in grade 10" M. 1986.
9. V. V. Afanasyev "Formation creative activity students in the process of solving mathematical problems "Yaroslavl 1996.
10. PI Altynov “Tests. Geometry 7 - 9 grades. " M. 1998.
11. Newspaper "Mathematics" 17/1996.
12. Newspaper "Mathematics" 3/1997.
13. NP Antonov, M. Ya. Vygodsky, V. V Nikitin, AI Sankin "Collection of problems in elementary mathematics". M. 1963.
14. GV Dorofeev, MK Potapov, N. Kh. Rozov "A manual on mathematics". M. 1973
15. AI Shchetnikov “The Pythagorean doctrine of number and magnitude”. Novosibirsk 1997.
16. “Real numbers. Irrational expressions "Grade 8. Tomsk University Press. Tomsk - 1997.
17. M.S. Atanasyan "Geometry" grade 7-9. M: Enlightenment, 1991
18.www.moy pifagor.narod.ru /
19.http: //www.zaitseva-irina.ru/html/f1103454849.html
20.http: //ru.wikipedia.org/wiki/Pythagoras_Teorem
21.http: //th-pif.narod.ru/history.htm
The history of the Pythagorean theorem goes back several millennia. A statement that was known long before the birth of the Greek mathematician. However, the Pythagorean theorem, the history of creation and its proof are associated for the majority with this scientist. According to some sources, the reason for this was the first proof of the theorem, which was given by Pythagoras. However, some researchers refute this fact.
Music and logic
Before telling how the history of the Pythagorean theorem developed, let us briefly dwell on the biography of the mathematician. He lived in the 6th century BC. The date of birth of Pythagoras is considered to be 570 BC. e., place - the island of Samos. Little is known for certain about the life of a scientist. Biographical data in ancient Greek sources are intertwined with sheer fiction. On the pages of treatises, he appears as a great sage, excellently commanding the word and the ability to convince. By the way, this is why the Greek mathematician was nicknamed Pythagoras, that is, "persuasive speech." According to another version, the birth of the future sage was predicted by the Pythia. The father named the boy Pythagoras in her honor.
The sage learned from the great minds of the day. Among the teachers of the young Pythagoras are Hermodamantus and Therekides of Syros. The first instilled in him a love of music, the second taught him philosophy. Both of these sciences will remain the focus of a scientist's attention throughout his life.
30 years of training
According to one version, being an inquisitive young man, Pythagoras left his homeland. He went to Egypt to seek knowledge, where he stayed, according to various sources, from 11 to 22 years, and then was captured and sent to Babylon. Pythagoras was able to benefit from his position. For 12 years he studied mathematics, geometry and magic in ancient state... Pythagoras returned to Samos only at the age of 56. The tyrant Polycrates ruled here at that time. Pythagoras could not accept such political system and soon went to the south of Italy, where the Greek colony of Croton was located.
Today it is impossible to say for sure whether Pythagoras was in Egypt and Babylon. Perhaps he left Samos later and went directly to Croton.
Pythagoreans
The history of the Pythagorean theorem is associated with the development of the school created by the Greek philosopher. This religious and ethical brotherhood preached the observance of a special way of life, studied arithmetic, geometry and astronomy, and studied the philosophical and mystical side of numbers.
All the discoveries of the students of the Greek mathematician were attributed to him. However, the history of the origin of the Pythagorean theorem is associated by ancient biographers only with the philosopher himself. It is assumed that he passed on the knowledge gained in Babylon and Egypt to the Greeks. There is also a version that he really discovered the theorem on the ratios of legs and hypotenuse, not knowing about the achievements of other peoples.
Pythagoras' theorem: history of discovery
Some ancient Greek sources describe the joy of Pythagoras when he managed to prove the theorem. In honor of such an event, he ordered a sacrifice to the gods in the form of hundreds of bulls and made a feast. Some scholars, however, point to the impossibility of such an act due to the peculiarities of the views of the Pythagoreans.
It is believed that in the treatise "Beginnings", created by Euclid, the author provides a proof of the theorem, the author of which was the great Greek mathematician. However, not everyone supported this point of view. For example, the ancient Neoplatonist philosopher Proclus pointed out that the author of the proof given in the Elements is Euclid himself.
Be that as it may, but the first who formulated the theorem, after all, was not Pythagoras.
Ancient Egypt and Babylon
The Pythagorean theorem, the history of the creation of which is considered in the article, according to the German mathematician Cantor, was known as early as 2300 BC. NS. in Egypt. The ancient inhabitants of the Nile Valley during the reign of Pharaoh Amenemhat I knew the equality 3 2 + 4 ² = 5 ². It is assumed that using triangles with sides 3, 4, and 5, the Egyptian "rope pulls" built right angles.
They knew the theorem of Pythagoras in Babylon. Clay tablets dating from 2000 BC and attributed to the time of the reign, an approximate calculation of the hypotenuse of a right-angled triangle was found.
India and China
The history of the Pythagorean theorem is also associated with the ancient civilizations of India and China. The treatise "Zhou-bi Xuan Jin" contains indications that (its sides are related as 3: 4: 5) was known in China as early as the XII century. BC e., and by the VI century. BC NS. mathematicians of this state knew general form theorems.
The construction of a right angle using the Egyptian triangle was also described in the Indian treatise "Sulva Sutra", dating back to the 7th-5th centuries. BC NS.
Thus, the history of the Pythagorean theorem at the time of the birth of the Greek mathematician and philosopher was already several hundred years old.
Proof
During its existence, the theorem has become one of the fundamental in geometry. The history of the proof of the Pythagorean theorem probably began with an examination of the equilateral. Squares are built on its hypotenuse and legs. The one that "grew" on the hypotenuse will consist of four triangles equal to the first. In this case, the squares on the legs consist of two such triangles. Simple graphic image clearly shows the validity of the statement formulated in the form of the famous theorem.
Another simple proof combines geometry with algebra. Four identical right-angled triangles with sides a, b, c are drawn so that they form two squares: an outer one with a side (a + b) and an inner one with a side c. In this case, the area of the smaller square will be equal to 2. The area of the large is calculated from the sum of the areas small square and all triangles (the area of a right-angled triangle, recall, is calculated by the formula (a * b) / 2), that is, with 2 + 4 * ((a * b) / 2), which is equal to 2 + 2av. The area of a large square can be calculated in another way - as the product of two sides, that is, (a + b) 2, which is equal to a 2 + 2av + b 2. It turns out:
a 2 + 2av + b 2 = c 2 + 2av,
a 2 + b 2 = c 2.
There are many known proofs of this theorem. Euclid, Indian scientists, and Leonardo da Vinci also worked on them. Often the ancient sages cited drawings, examples of which are located above, and did not accompany them with any explanations, except for the note "Look!" The simplicity of the geometric proof, subject to some knowledge, did not require comments.
The history of the Pythagorean theorem, summarized in the article, debunks the myth of its origin. However, it is difficult even to imagine that the name of the great Greek mathematician and philosopher would someday cease to be associated with her.
The fate of other theorems and problems is peculiar ... How can one explain, for example, such exceptional attention on the part of mathematicians and amateurs of mathematics to the Pythagorean theorem? Why were many of them not satisfied with the already known proofs, but found their own, bringing the number of proofs to several hundred in twenty-five comparatively foreseeable centuries?When it comes to the Pythagorean theorem, the unusual begins with its name. It is believed that Pythagoras was not the first to formulate it. It is also considered doubtful that he gave her proof. If Pythagoras is a real person (some even doubt this!), Then he lived, most likely, in the 6th-5th centuries. BC NS. He himself did not write anything, called himself a philosopher, which meant, in his understanding, "striving for wisdom", founded the Pythagorean Union, whose members were engaged in music, gymnastics, mathematics, physics and astronomy. Apparently, he was also an excellent orator, as evidenced by the following legend relating to his stay in the city of Crotone: “The first appearance of Pythagoras before the people in Crotone began with a speech to the young men, in which he was so strict, but at the same time so fascinating outlined the responsibilities of the young men, that the elders in the city asked not to leave them without instruction. In this second speech, he pointed to legality and purity of morals as the foundations of the family; in the next two he addressed children and women. The consequence of the last speech, in which he especially condemned luxury, was that thousands of precious dresses were delivered to the temple of Hera, for no woman dared to show herself in them on the street anymore ... ”Nevertheless, even in the second century AD, that is, after 700 years, they lived and worked quite real people, outstanding scientists, clearly under the influence of the Pythagorean union and with great respect for what, according to legend, Pythagoras created.
There is no doubt that interest in the theorem is also caused by the fact that it occupies one of the central places in mathematics, and by the satisfaction of the authors of the proofs who overcame the difficulties, about which the Roman poet Quintus Horace Flaccus, who lived before our era, spoke well: "It is difficult to express well-known facts." ...
Initially, the theorem established the relationship between the areas of squares built on the hypotenuse and legs of a right triangle:
.
Algebraic formulation:
In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
That is, denoting the length of the hypotenuse of the triangle through c, and the lengths of the legs through a and b: a 2 + b 2 = c 2. Both statements of the theorem are equivalent, but the second statement is more elementary, it does not require the concept of area. That is, the second statement can be checked without knowing anything about the area and measuring only the lengths of the sides of a right-angled triangle.
The converse theorem Pythagoras. For any triple of positive numbers a, b and c such that
a 2 + b 2 = c 2, there is a right-angled triangle with legs a and b and hypotenuse c.
Proof
At the moment, 367 proofs of this theorem have been recorded in the scientific literature. Probably the Pythagorean theorem is the only theorem with such an impressive number of proofs. This variety can be explained only by the fundamental meaning of the theorem for geometry.
Of course, conceptually all of them can be divided into a small number of classes. The most famous of them are: proofs by the area method, axiomatic and exotic proofs (for example, using differential equations).
Through similar triangles
The following proof of the algebraic formulation is the simplest of the proofs built directly from the axioms. In particular, it does not use the concept of the area of a figure.
Let ABC be a right-angled triangle with right angle C. Draw the height from C and denote its base by H. Triangle ACH is similar to triangle ABC in two angles. 
Likewise, triangle CBH is similar to ABC. Introducing the notation
we get 
What is the equivalent
Adding, we get
or 
Areas proof
The proofs below, despite their seeming simplicity, are not at all so simple. All of them use the properties of area, the proof of which is more difficult than the proof of the Pythagorean theorem itself.
Equal complementarity proof
1. Place four equal right-angled triangles as shown in the figure. 2. A quadrilateral with sides c is a square, since the sum of two acute angles is 90 °, and the unfolded angle is 180 °.
3. The area of the whole figure is, on the one hand, the area of a square with sides (a + b), and on the other hand, the sum of the areas of four triangles and an inner square.
Q.E.D.
Evidence through scattering
An example of one of such proofs is shown in the drawing on the right, where a square built on the hypotenuse is transformed by permutation into two squares built on the legs.
Euclid's proof
The idea behind Euclid's proof is as follows: let's try to prove that half of the area of the square built on the hypotenuse is equal to the sum of the halves of the areas of the squares built on the legs, and then the areas of the large and two small squares are equal. Consider the drawing on the left. On it, we built squares on the sides of a right-angled triangle and drawn a ray s from the vertex of the right angle C perpendicular to the hypotenuse AB, it cuts the square ABIK, built on the hypotenuse, into two rectangles - BHJI and HAKJ, respectively. It turns out that the areas of these rectangles are exactly equal to the areas of the squares built on the corresponding legs. Let's try to prove that the area of the square DECA is equal to the area of the rectangle AHJK To do this, let's use an auxiliary observation: The area of a triangle with the same height and base as this rectangle is equal to half the area of the given rectangle. This is a consequence of the definition of the area of a triangle as half of the product of the base and the height. From this observation it follows that the area of the triangle ACK is equal to the area of the triangle AHK (not shown in the figure), which, in turn, is equal to half the area of the rectangle AHJK. Let us now prove that the area of the triangle ACK is also equal to half the area of the square DECA. The only thing that needs to be done for this is to prove the equality of the triangles ACK and BDA (since the area of the triangle BDA is equal to half the area of the square according to the above property). Equality is obvious, the triangles are equal on two sides and the angle between them. Namely - AB = AK, AD = AC - the equality of the angles CAK and BAD is easy to prove by the method of motion: we rotate the triangle CAK 90 ° counterclockwise, then it is obvious that the corresponding sides of the two triangles under consideration will coincide (since the angle at the apex of the square is 90 °). The reasoning about the equality of the areas of the square BCFG and the rectangle BHJI is completely analogous. Thus, we have proved that the area of the square built on the hypotenuse is the sum of the areas of the squares built on the legs. Proof of Leonardo da Vinci
The main elements of the proof are symmetry and motion.
Consider the drawing, as can be seen from the symmetry, the segment CI cuts the square ABHJ into two identical parts (since the triangles ABC and JHI are equal in construction). By rotating it 90 degrees counterclockwise, we see that the shaded figures CAJI and GDAB are equal. Now it is clear that the area of the shaded figure is equal to the sum of the halves of the areas of the squares built on the legs and the area of the original triangle. On the other hand, it is equal to half the area of the square built on the hypotenuse plus the area of the original triangle. The final step in the proof is left to the reader.